So, to clarify my question a bit, were there East Germans, whether a part of the government, or part of a civilian movement, that wanted to see the end of communism within East Germany, but also did not want to reunify with West Germany?
Most people seem to correlate the fall of The Berlin Wall as the beginning of the end for East Germany as a nation, but I've always found that history is never so cut and dry.
Reintegrating East Germany into West Germany would seemingly be a massive undertaking. From infrastructure, to merging armed forces, to the political enfranchisement of roughly 16 million East Germans in the West German political system, the task seems very daunting.
So, was there ever an effort to just not bother reunifying, and leave East Germany as a separate, sovereign, non-communist state? If so, did it have any traction?
Yes, actually. I'm going to start with some background as a way to set the stage and introduce the unholy amount of abbreviations we'll have to familiarise ourselves with. As a caveat, I'll stick to original German abbreviations, as that seems to be common practice, at least for party names.
The SEP (Sozialistische Einheitspartei Deutschlands, Eng: 'Socialist Unity Party of Germany') ruled the DDR (Deutsche Demokratische Republik, Eng: German Democratic Republic, aka 'East Germany') as a de facto one-party state. (There were, in fact, other parties permitted to take part in what might appear to be some sort of parliamentary system like in Western liberal democracies, but these all took part in the 'National Front' organisation which was headed, of course, by the SEP. This was actually a common arrangement across the Eastern European communist nations. See for example the 'National Front of Czechs and Slovaks' in communist Czechoslovakia.)
I bring this up because there was a key transition step between the end of one-party rule and reunification which gets overlooked a lot. The DDR did in fact have one free election in 1990, wherein pro-reunification parties won a decisive majority by the coalition between the Alliance for Germany and the SPD (192 and 88 seats respectively, out of a total of 400 in the DDR's legislature, the Volkskammer.) It's the party in third place, with 16.4% of the national vote and 66 seats, that we are interested in.
This was the PDS (Partei des Demokratischen Sozialismus, Eng: 'Party of Democratic Socialism'), which was in fact the successor/continuation of the SED. By the time of the election, Erich Honecker (who had led the country from 1971-1989) and his short-lived successor Ergon Krentz had been expelled by the party, which was radically rebranding itself. The quote "Wir brechen unwiderruflich mit dem Stalinismus als System" (Eng: "We break irrevocably with Stalinism as a system") by East German philosopher, Michael Schumann, which he made at an SED extraordinary Party Conference on 16 December 1989, has become quite famous.
The PDS's platform seems to have broadly to been to maintain the socialist basis of the East German economy and social benefit programmes, while also accepting the new democratic system. One campaign poster I've managed to track down here which they used in 1990 read: "Demokratische Freiheit für alle Soziale Sicherheit für jeden für ein selbstbewußtes Berlin" (Eng: "Democratic freedom for all Social security for everyone For a self‐confident Berlin").
I hope this answer didn't disappoint you. I would consider the reformed PDS as "non-communist" insofar as they had ostensibly abandoned Marxism-Leninism and authoritarian rule, but they were undeniably still socialist.